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Theorem coeq12d 5167
Description: Equality deduction for composition of two classes. (Contributed by FL, 7-Jun-2012.)
Hypotheses
Ref Expression
coeq12d.1  |-  ( ph  ->  A  =  B )
coeq12d.2  |-  ( ph  ->  C  =  D )
Assertion
Ref Expression
coeq12d  |-  ( ph  ->  ( A  o.  C
)  =  ( B  o.  D ) )

Proof of Theorem coeq12d
StepHypRef Expression
1 coeq12d.1 . . 3  |-  ( ph  ->  A  =  B )
21coeq1d 5164 . 2  |-  ( ph  ->  ( A  o.  C
)  =  ( B  o.  C ) )
3 coeq12d.2 . . 3  |-  ( ph  ->  C  =  D )
43coeq2d 5165 . 2  |-  ( ph  ->  ( B  o.  C
)  =  ( B  o.  D ) )
52, 4eqtrd 2508 1  |-  ( ph  ->  ( A  o.  C
)  =  ( B  o.  D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    o. ccom 5003
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-in 3483  df-ss 3490  df-br 4448  df-opab 4506  df-co 5008
This theorem is referenced by:  xpcoid  5548  dfac12lem1  8524  dfac12r  8527  imasval  14769  cofuval  15112  cofu2nd  15115  cofuval2  15117  cofuass  15119  cofurid  15121  setcco  15271  isdir  15722  evl1fval  18175  znval  18379  znle2  18399  mdetfval  18895  mdetdiaglem  18907  ust0  20549  trust  20559  metustexhalfOLD  20893  metustexhalf  20894  isngp  20943  ngppropd  20978  tngval  20980  tngngp2  20993  imsval  25364  opsqrlem3  26834  hmopidmch  26845  hmopidmpj  26846  pjidmco  26873  dfpjop  26874  zhmnrg  27699  dfrtrcl2  28822  istendo  35773  tendoco2  35781  tendoidcl  35782  tendococl  35785  tendoplcbv  35788  tendopl2  35790  tendoplco2  35792  tendodi1  35797  tendodi2  35798  tendo0co2  35801  tendoicl  35809  erngplus2  35817  erngplus2-rN  35825  cdlemk55u1  35978  cdlemk55u  35979  dvaplusgv  36023  dvhopvadd  36107  dvhlveclem  36122  dvhopaddN  36128  dicvaddcl  36204  dihopelvalcpre  36262  trrelind  37005  trficl  37006  trrelsuperreldg  37010  trclub  37011  trclubg  37012
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